In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space.
http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf
After which they prove that their exists those of order 2 and 3. After breaking my head on it for a bit i can't seem to wrap my head around the prove.
What is the second order vector they prove that exists in that case. I would already be quite happy with a concrete example.